It is not obligatory to wash the mouth after eating or drinking, whether you want to pray or read the Quran. But it is recommended for you to do that especially if what he ate or drank was at all greasy.

It is not permissible to throw leftover food and drink that can be used down the drain or to put them out with the trash. With regard to food that is not good or that has gone off, and drinks that cannot be used, such as leftover tea and coffee, there is nothing wrong with throwing them out with the trash or into drains.